SAMPLE MIDTERMS
MATH 55
Sample 1.
1. Here is the definition of a strictly increasing function f : R → R:
∀x, y ∈ R (x < y → f (x) < f (y)).
Write in the similar way the statement that f is not strictly increasing.
2. Which of the following sets are infinitely countable
(a) Z × Z,
(b) R − Z,
(c) the set of all irrational real numbers?
Justify your answer.
3. Find the binary and hexadecimal expansions for 500.
4. Solve the following system of congruence equations
x ≡ 3(
mod 5)
x ≡ 4(
mod 7)
x ≡ 2( mod 9).
5. Prove by induction the identity
n
X
n(n + 1)(n + 2)
i(i + 1) =
.
3
i=1
Sample 2.
1. Prove that for three sets A, B and C, |A| = |B| and |B| = |C| implies |A| = |C|.
2. A sequence an is strictly increasing if for all n, an+1 > an . Let an be given be
given recursively
a1 = 1, an = a2n−1 + an−1 for n = 2, 3, 4, . . .
(a) Write first five terms of the sequence.
(b) Prove that an is strictly increasing.
3. Find gcd(2012, 344) using Euclidean algorithm.
4. Find all prime p such that p2 + 14 is also prime. (Hint: use arithmetic modulo
3.)
5. Let An = {nk | k ∈ Z}. Find
∞
\
An .
n=1
Sample 3
1
2
SAMPLE MIDTERMS MATH 55
1. Let f : A → B be a surjective function. Prove that |B| ≤ |A|.
2. Prove that the number is divisible by 9 iff the sum of its digits is divisible by 9.
3. Using strong induction prove that any positive integer can be written as a sum
of distinct Fibonacci numbers.
4. Let P (n) denote the predicate n2 + n + 41 is prime.
(a) Find the truth values of P (1), P (2), P (5).
(b) Find n such that P (n) is false.
5. Write in pseudocode the procedure which finds a primitive root modulo a prime
p.